The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

4.1. Bound and scattering state equations 121
where J and E denote the quantities in the Blatt-Biedenharn parametrization.
The bound state is obtained from the homogeneous part of eq. (4.5) and obeys
where s = j = 1, [ = [' = 0,2 and Ed denotes the bound-state energy.
( 4.10)
The potential (4.1)-(4.3) is only meaningful for momenta below a certain scale. This is obvious
both from a conceptual and a practical point of view. Indeed, since we are working within
the effective field theory approach, only the low-momentum matrix elements can be derived
systematicaHy. Our perturbative expansion far the potential breaks down for momenta of the
nucleons comparable with the scale Ax. Also from the practical point of view, the expressions
(4.1)-(4.3) are, clearly, not acceptable for large values of the moment um transfer q. In particular,
since the potential grows for large momenta, it leads to ultraviolet divergences in the LS equation
(4.5). As it is appropriate in effective theory, we regularize the potential. This is done in the
following way:
(4.11)
where fR(iJ) is a regulator function chosen in harmony with the underlying symmetries. In what
follows, we work with two different regulator functions,
ftarp(p)
f �xpon (p)
B(A2 _ p2) ,
exp( _p2n / A 2n) ,
( 4.12)
(4.13)
with n = 2,3, ... , p = Ipl and p' = Ip'l. Such a regularization of the effective potential is also
consistent with the discussion in chapter 2. We must introduce a cut-off in order to exclude
the high-momentum intermediate states in the LS equation, that can not be described correctly
within the low-energy effective theory.
In eqs. (4.12), (4.13) we show the two different choices of the regulator function. The sharp cut-off
is most appropriate for comparison with realistic phenomenological potentials. To enable such a
comparison one first needs to integrate out the high-momentum components from the realistic
potentials, since the effective one is defined only for momenta p, p' < A. Eliminating the highmoment
um components can be performed via the unitary transformation, as described in section
2.3. We already performed that step but leave the discussion of the results to a furt her study. The
sharp cut-off regularization scheme is suitable for deriving the phase shifts but leads to troubles
when calculating some deuteron properties. This is because discontinuities in the moment um
space are introduced at p, p' = A. In that case one should use a smooth regularization like the
exponential one given in eq. (4.13). For very large integers n the exponential cut-off approximates
the sharp one. Throughout, we work with n = 2. To the order we are working, the choice n = 1
has to be excluded since the terms of order p2,p /2 would be modified. For n = 2, the error we
make is beyond the accuracy of the order we are calculating. We would like to point out that the
low-energy observables should not be sensitive to the choice of the regulating function fR as weH
as to an exact value of the cut-off. We will also confirm this statement by explicit calculations.
The remaining (weak) dependence of the observables on the choice of fR and on the value of A
can be systematically removed by including the higher order terms [141]. In the following seetions
we will discuss these uncertainties in more details.